In this paper an approximation theory is provided for the solutions of infinite dimensional Algebraic Riccati Equations, which in particular includes convergence of the approximating Riccati operators as well as convergence of the approximating gain operators. The main features which distinguish this paper from other work existing in the literature of Riccati approximation theory are: (i) the original free system only generates a strongly continuous semigroup; (ii) the input (control) operator is generally unbounded; and (iii) no “smoothing” hypothesis on the observation operator is assumed. The abstract theory is illustrated by several examples arising in boundary control problems for wave and plate equations.